What allows us to assume spacetime is flat when no normal matter is present? Dark matter causes a bend in spacetime.  We see this through gravitational lensing. But what allows us to assume spacetime is flat when no normal matter is present? Why can't we say that dark matter is just the natural curvature of spacetime?  Flat seems like a special case compared to the rest of possible geometries. 
 A: We assume that most of space is flat for pretty much the same reason that we assume that an expanse of glass is flat when the images we see through it are undistorted. In the case of astronomy, the images seen through the "glass" of space consists mostly of distant galaxies and galaxy clusters.
In sharp contrast, dark matter distorts images over just a tiny part of space. It acts much like a magnifying glass, producing magnified and weirdly distorted images of even more distant galaxies. The effect is called an Einstein lens, and yes, it can even be used (with computer image processing) as a sort of universe-sized telescope.
A: It's not an assumption, it's a fundamental aspect of general relativity (GR), which has been shown to be a very accurate description of many observable phenomena.  The keystone of GR is Einstein's Field Equations which we can write as,  
$$\textrm{curvature of spacetime} = \textrm{energy-density}$$
The left-hand side of the equation is more precisely expressed with specific (complex) measures of curvature (like metrics and curvature tensors etc), while the right-hand side is usually given by a stress-energy tensor which includes all types of matter --- and not even only 'matter' in the classical sense, but also radiation, etc.  So, in effect, general relativity says that curvature and energy (of any form) are inextricable --- and thats the core of how we view spacetime and cosmology.
Edit:
There's actually another way to think about your question -- which is historically, in terms of how the dark matter theory developed.  Basically, we observed gravitational forces (spacetime curvature) which couldn't be associated with known forms of matter, and so Dark Matter (DM) was postulated as the source of that additional curvature.  So, in an indirect sort of way, dark matter is just the spacetime curvature that we've observed... but, both because of GR, and because of how we observe DM to be distributed throughout the universe, we don't think this is causally 'just the natural' property of spacetime.
Please comment if I'm misinterpreting the crux of the question (e.g. what you mean by 'normal' matter).
A: It would be possible for a natural curvature of spacetime to appear as dark matter. Dark energy can be described in this way. Putting dark energy on the right hand side of the Einstein equation interprets it as matter/energy while putting it on the left hand side interprets it as a geometrical property of spacetime.
The problem with dark matter is that it is uneven. For example the observations of the Bullet Cluster suggest there are concentrations of dark matter that (before the collision) corresponded to the visible matter density.
When describing the expansion of the universe as a whole we could model dark matter as a geometrical property, just like dark energy. However this wouldn't help us explain the observed unevenness in the distribution of dark matter. That can only be explained by putting the dark matter on the right hand side of the Einstein equation i.e. as some form of weakly interacting matter scattered around the universe.
One last point, the flat geometry actually seems most natural to most physicists. That's because if the universe is curved that curvature changes (increases) with time. If the universe was even slightly curved just after the Big Bang then it would be massively curved now - 13.8 billion years later. However if the universe started flat it stays flat. Since the universe obviously isn't massively curved right now the simplest explanation is that it started flat and therefore has remained flat.
A: 
What allows us to assume spacetime is flat when no normal matter is present?

Nothing. However people do make this assumption, and then propose exotic dark matter particles to explain gravitational anomalies such as flat galactic rotation curves. This is despite Einstein saying "the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy". A gravitational field is a region of space where the energy-density is higher than free space. This spatial energy has a mass equivalence and a gravitational effect, but it isn't made of particles. Instead it's where space is "neither homogeneous nor isotropic" in a non-linear fashion. This is the same thing as curved spacetime, see Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime. Also see Inhomogeneous and interacting vacuum energy.  

Are there any other theories that explains movements of galaxies, universe clustering and other observations (that lead us to assume dark matter existence) as an exotic states of space-time?

Maybe f(R) gravity is what you're looking for, but the "exotic states of space-time" has me struggling a bit. A gravitational field is inhomogeneous space = curved spacetime, but it isn't exotic.   

Dark matter causes a bend in spacetime. We see this through gravitational lensing. 

See the abstract for the first inhomogeneous vacuum paper: "The strong similarities between the light propagation in a curved spacetime and that in a medium with graded refractive index are found. It is pointed out that a curved spacetime is equivalent to an inhomogeneous vacuum for light propagation. The corresponding graded refractive index of the vacuum in a static spherically symmetrical gravitational field is derived. This result provides a simple and convenient way to analyse the gravitational lensing in astrophysics."

But what allows us to assume spacetime is flat when no normal matter is present? Why can't we say that dark matter is just the natural curvature of spacetime? Flat seems like a special case compared to the rest of possible geometries.

I share your sentiment. Space is dark, it has its vacuum energy which has a mass equivalence, and there's a lot of it about. And for myself I think conservation of energy means space has to be inhomogeneous. The raisin-cake analogy says space expands between the galaxies but not within. So every galaxy is surely embedded in region of space with a higher energy-density than intergalactic space, such that it's surrounded by a halo of inhomogeneous space. And according to Einstein, inhomogeneous space is what a gravitational field is.  
